Download Propulsion for Small Electric Aircraft

AA241X April 2007
Propulsion for Small Electric Aircraft
This section describes a few of the basic considerations needed to estimate the
performance of small electric aircraft propulsion systems. An efficient aircraft
requires a careful matching of motor, propeller, and airframe -- a matching that is
sometimes difficult when the design is highly constrained (as in AA241X).
Computer programs such as some of the applets here and specialized programs such
as MotoCalc (www.motocalc.com) provide useful ways to estimate system
performance, but can sometime be too helpful, eliminating the need to understand
why systems behave as they do.
The following notes are intended to introduce you to some of the physics underlying
such calculations. Enter data in the applets below and click "Compute" to update.
Drag the mouse on the plot to zoom in; shift-click to auto-scale.
Electric Motor Performance
Electric motors are characterized by their weight, design voltage, and maximum
power output, but also by several other parameters that determine how the
efficiency and power vary with RPM. If gearboxes are available, the optimal RPM of
the propeller and motor can be made to match, but for direct drive systems, the
matching of these characteristics is especially important.
One can compute many useful electric motor characteristics from the 3 basic
equations shown below. In this analysis, the motor performance is characterized by
3 basic constants:
Kv relates the motor speed (RPM) to the voltage applied to the windings. This
characterizes how the motor's back emf changes with speed. For many motors, this
relation is roughly linear so we can write:
RPM = Kv (V - I R)
Where V is the applied voltage, I is the current, and R is the armature resistance.
Kv is often specified for a particular motor, or it can be measured by testing the
motor and measuring the unloaded RPM and current at several voltages.
R is also often published or can be measured directly on the motor.
The motor torque is related to the applied current (since this sets the strength of the
magnetic fields generated by the motor windings). The expression is:
Q = Kt (I - I0)
Where Q is the torque and I0 is the no-load current at the specified voltage. I0 is
also sometimes specified in motor catalogs or can be measured directly on a given
motor.
Noting that Q RPM is proportional to output power and that (V-IR)*(I-I0) is the input
power minus losses, we see there is a direct relationship between Kt and Kv. With Kt
in RPM/volt and Kv in in-oz (as in MotoCalc), the expression is:
Kv Kt = 1352.
One could do this in many unit systems, but the idea is that only one of these motor
parameters (usually Kv) needs to be specified.
The net result is that when the motor is characterized by Kv, I0, and R, one may
compute the output power, input power (I V), torque, and efficiency. From the
equations above. With proper unit conversions, this is what is done in the applet in
these notes.
Kt = 1352./Kv;
I = (volts - RPM/Kv)/R;
Q = Kt*(I-I0); // Q in in oz if Kt defined as in Motocalc
QinNm = Q*.00706; // Conversion to N m
Pin = volts*I; // watts
Pout = QinNm*RPM*2.*pi/60.; // watts
eta = Pout/Pin; // Motor Efficiency
Propeller Performance
The analysis of propellers ranges from very simple momentum arguments to very
sophisticated computer programs that can tax even modern supercomputers. In the
following discussion, we consider a few approaches to propeller analysis.
Maximum Efficiency
The first step in the study of propeller performance is an inviscid momentum analysis
with lots of simplifying assumptions. The thrust produced by an annular region of the
propeller at radius r may be estimated by considering the rate of momentum change
through the propeller at this station:
dT = ρ (U+u) (2π r dr) 2u
From a vortex model perspective, the blades are generating lift due to their
circulation:
dT = Nblades ρ Γ ( ω r) dr
The induced angle generates some induced drag on the blades so that at each
station, r, the incremental torque is given by:
dQ = Nblades ρ Γ (U+u) r dr
Using the second expression to compute the blade circulation, we have:
Γ = (U+u) (4π) u / (ω Nblades)
If the blades generate a uniform (constant with radius, r) increase in velocity
through the propeller plane (u) then the blade circulation is also constant and we can
integrate the expressions above to find:
T = 2 ρ (U+u) u A
Q = 2 ρ (U+u) 2 u A/ω
where: A = π R2, the disk area.
This means that ω Q = (U+u) T
So the efficiency is: η = Pout / Pin = T U / ω Q = U / (U+u).
Now the expression for T can be used to relate u to the other parameters. Solving
the quadratic: 0 = u2 + Uu - T/2ρA
2u/U = sqrt( 1 + 2T/ρAU2) - 1
Note that when the propeller is lightly loaded (2T/ρAU2 << 1) the above simplifies
to: u/U = T/2ρAU2
So: η = 1 / (1 + T/2ρAU2)
This expression illustrates a few things, even though we have ignored many features
such as nonuniform u, viscous drag, propeller swirl, and others. First, the basic
momentum efficiency limits propeller performance at low speeds. For example, if we
need to generate a thrust of just 1/2 oz at 10 ft/sec with a 3 in propeller, the inflow
velocity ratio is u/U = .76 and the efficiency is no more than 57%. Other effects will
reduce this further.
If the units above bothered you, then good. Propeller performance is generally
computed in dimensionless terms, with the following (rather odd)
nondimensionalization most common:
• CT = T/ ρ n2 D4
• C Q = Torque / ρ n2 D5
• C P = Power_In / ρ n3 D5
• J = advance ratio = U / n D
where n = revolutions per second, and D = diameter.
So in dimensionless terms: 2u/U = sqrt( 1 + 8C T/πJ2) - 1
or: η = 2 / (1+sqrt( 1 + 8CT/πJ2)).
Propeller Thrust
Propeller thrust is specified in the above expressions and often we are interested in
the available thrust from a specific propeller. This can be done from a more firstprinciples analysis, or based on some measured data.
One additional parameter that is useful in describing a propeller geometry is the
pitch. If a propeller were designed so that its sections would follow streamlines,
generating no lift at a certain speed (the pitch speed), then we would define the
distance that it travelled forward in each revolution as the pitch:
pitch = Ups / (ω/2π) = Ups / n
The advance ratio at the pitch speed is given by:
Jps = Ups / n D = pitch / D.
So if we have a propeller with a pitch/D of 50/82, it will generate no thrust at J =
.61. We can also estimate the torque at this condition, which is due to viscous drag.
From the expressions at the link below, the result is:
dQ = Nblades ρ/2 ω sqrt(Ups2+ ω2r2) Cd c r2 dr
which can be easily integrated with an assumed distribution of chord and Cd.
At the static condition, the angle of attack distribution is not so simple and the
assumption of constant inflow, u, is not so good. There are a few choices here: one
can model the actual incidence, chord, and Cd distributions and iterate to find the
inflow distribution, thrust, and torque. One can assume the inflow is constant
anyway (probably OK for rough performance estimates). Or one can measure the
static thrust and infer the average inflow. MotoCalc appears to use some
approximate curves of CT and CQ vs. J and modify these based on measured static
thrust -- fine for rough performance estimates, but not for propeller design.
Additional details on these calculations and the theory behind vortex-momentum
theory of propeller analysis is provided at this link.
Combined Motor and Propeller Performance
Combining the motor and propeller analysis allows us to compute the variation of
thrust, motor current, and efficiency of the propulsion system as shown in the applet
below. In this case, the program iterates on RPM at each speed to find the propeller
speed at which required torque is equal to that provided by the motor. It may be
useful to fit this data in order to take the next step: matching propulsion system
characteristics to your airframe characteristics.